Coupling model, relaxation times

Ngai and Phillies(61) extend the Ngai-Rendell coupling model(62) to treat probe diffusion and polymer dynamics in polymer solutions. This is not an experimental paper it forces extant experimental data to confront a particular theoretical model, which in the paper was extensively reconstructed to treat the particular experimental methods under consideration. Ngai and Phillies consider zero-shear viscosities and optical probe diffusion spectra for HPC solutions, extracting from them the Ngai-Rendell model relaxation time tq and coupling exponent n. Optical probe spectra and other measurements were used to obtain n in four independent ways, namely from r](c) and from the concentration, time, and wave-vector dependences of g q,t). The four paths from t] and lead to consis-... 

According to the coupling model, relaxation parameters e.g. ordering parameters or mechanisms in the KAHR model) do not exhibit simple exponential decay. Rather, the physics of complex systems requires cooperativity among the relaxing species which acts to slow the simple constant relaxation rate into a time-dependent rate. Then one finds that the relevant variable initially decays at a constant rate, PFq = Tq S but at times larger than a characteristic time the rate becomes... 

For example, if the molecular structure of one or both members of the RP is unknown, the hyperfine coupling constants and -factors can be measured from the spectrum and used to characterize them, in a fashion similar to steady-state EPR. Sometimes there is a marked difference in spin relaxation times between two radicals, and this can be measured by collecting the time dependence of the CIDEP signal and fitting it to a kinetic model using modified Bloch equations [64]. 

Often the electronic spin states are not stationary with respect to the Mossbauer time scale but fluctuate and show transitions due to coupling to the vibrational states of the chemical environment (the lattice vibrations or phonons). The rate l/Tj of this spin-lattice relaxation depends among other variables on temperature and energy splitting (see also Appendix H). Alternatively, spin transitions can be caused by spin-spin interactions with rates 1/T2 that depend on the distance between the paramagnetic centers. In densely packed solids of inorganic compounds or concentrated solutions, the spin-spin relaxation may dominate the total spin relaxation 1/r = l/Ti + 1/+2 [104]. Whenever the relaxation time is comparable to the nuclear Larmor frequency S)A/h) or the rate of the nuclear decay ( 10 s ), the stationary solutions above do not apply and a dynamic model has to be invoked... 

The longest mode (p=l) should be identical to the motion of the chain. The fundamental correctness of the model for dilute solutions has been shown by Ferry [74], Ferry and co-workers [39,75] have shown that,in concentrated solutions, the formation of a polymeric network leads to a shift of the characteristic relaxation time A,0 (X0=l/ ycrit i.e. the critical shear rate where r becomes a function of y). It has been proposed that this time constant is related to the motion of the polymeric chain between two coupling points. 

The concept of a T2 cut-off that partitions the relaxation time distribution between the pores which can be displaced and those that cannot does not always apply. An exception is when there is significant diffusional coupling between the micropores that retain water at a high capillary pressure and the macropores in close proximity to the microporous system [26, 27]. A spectral BVI model or a forward model has been suggested to interpret these systems [30, 31, 53]. 

Kinetic schemes involving sequential and coupled reactions, where the reactions are either first-order or pseudo-first order, lead to expressions for concentration changes with time that can be modeled as a sum of exponential functions where each of the exponential functions has a specific relaxation time. More complex equations have to be derived for bimolecular reactions where the concentrations of reactants are similar.19,20 However, the rate law is always related to the association and dissociation processes, and these processes cannot be uncoupled when measuring a relaxation process. 

To more fully appreciate the equilibrium models, like SCRF theories, and their usefulness and limitations for dynamics calculations we must consider three relevant times, the solvent relaxation time, the characteristic time for solute nuclear motion in the absence of coupling to the solvent, and the characteristic time scale of electronic motion. We treat each of these in turn. 

This approximation requires that cos. This behavior in fact follows from a Debye dielectric continuum model of the solvent when it is coupled to the solute nuclear motion [21,22] and then xs would be proportional to the longitudinal dielectric relaxation time of the solvent indeed, in the context of time dependent fluorescence (TDF), the Debye model leads to such an exponential dependence of the analogue... 

Inner slip, between the solid wall and an adsorbed film, will also influence the surface-liquid boundary conditions and have important effects on stress propagation from the liquid to the solid substrate. Linked to this concept, especially on a biomolecular level, is the concept of stochastic coupling. At the molecular level, small fluctuations about the ensemble average could affect the interfacial dynamics and lead to large shifts in the detectable boundary condition. One of our main interests in this area is to study the relaxation time of interfacial bonds using slip models. Stochastic boundary conditions could also prove to be all but necessary in modeling the behavior and interactions of biomolecules at surfaces, especially with the proliferation of microfluidic chemical devices and the importance of studying small scales. 

Relaxation times for water filling the pores of an NaX specimen have been fitted to a model with the following assumptions (a) coupling, as above, of molecular diffusion and rotation (b) the median jump time r is governed by a free volume law (allows the curvature in the plots of jump rate, (3r) x vs. 10S/T in Figure 5), and (c) a broad distribution of correlation times (allows a better fit to the data, accounts for an apparent two-phase behavior in T2 (31, 39), and is reasonable in terms of the previous discussion of Pi(f) and r). 

Hereafter we put /ig = 1. Below we express our results in terms of the statistical properties (correlators) of the environment s noise, X(t). Depending on the physical situation at hand, one can choose to model the environment via a bath of harmonic oscillators [6, 3]. In this case the generalized coordinate of the reservoir is defined as X = ]T)Awhere xi are the coordinate operators of the oscillators and Aj are the respective couplings. Eq. 2 is then referred to as the spin-boson Hamiltonian [8]. Another example of a reservoir could be a spin bath [11] 5. However, in our analysis below we do not specify the type of the environment. We will only assume that the reservoir gives rise to markovian evolution on the time scales of interest. More specifically, the evolution is markovian at time scales longer than a certain characteristic time rc, determined by the environment 6. We assume that rc is shorter than the dissipative time scales introduced by the environment, such as the dephasing or relaxation times and the inverse Lamb shift (the scale of the shortest of which we denote as Tdiss, tc [Pg.14]

F. Ingrosso, B. Mennucci and J. Tomasi, Quantum mechanical calculations coupled with a dynamical continuum model for the description of dielectric relaxation time dependent Stokes shift of coumarin Cl53 in polar solvents, J. Mol. Liq., 108 (2003) 21 -6. 

Molecules with small spin have also been discussed. For example, time-resolved magnetization measurements were performed on a spin 1/2 molecular complex, so-called V15 [81]. Despite the absence of a barrier, magnetic hysteresis is observed over a time scale of several seconds. A detailed analysis in terms of a dissipative two-level model has been given, in which fluctuations and splittings are of the same energy. Spin-phonon coupling leads to long relaxation times and to a particular butterfly hysteresis loop [58, 82],... 

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